Finite Group Databasebeta

← #37550351

Subgroups — #37550351

Order 23168.

738 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 198 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26042358C211yesmaximal, characteristic, complemented
2#41×#10624382C211yesmaximal, characteristic, complemented
3#26042277C213yesmaximal, complemented
4#26042276C213yesmaximal, complemented
5D181⋊#104C213yesmaximal, complemented
6C2×#19153880C213yesmaximal, complemented
7#26042114C211yesmaximal, characteristic
8#19153904—21no—
9#19153903—23no—
10C2×#15668459C2213yescomplemented
11D181⋊M16C2213yescomplemented
12Q8×#10624382—21no—
13D4×#10624382—23no—
14#19153895C2213yescomplemented
15C2×C4×#10624382C2213yescomplemented
16#19153881C2211yescharacteristic, complemented
17#19153881C2213yescomplemented
18#19153880C2213yescomplemented
19#19153880C2216yescomplemented
20#19153880C2213yescomplemented
21#19153863—21no—
22#19153863C411yescharacteristic, complemented
23#19153863C413yescomplemented
24#19153861—23no—
25#19153861C413yescomplemented
26C2×#15668448—23no—
27C2×#15668448C2213yes—
28#41×D181—21no—
29#41×D181C2211yescharacteristic
30#41×D181C2213yes—
31H(5,𝔽2)×C181C411yescharacteristic, complemented
32C22×#10624382—23no—
33#15668459—23no—
34#15668459C2311yescharacteristic
35#15668459C2313yes—
36#15668458—23no—
37D181⋊C8—23no—
38C2×#13089685—23no—
39C4⋊#10624382—23no—
40C22⋊#10624382—23no—
41C4×#10624382—23no—
42C4×#10624382C2311yescharacteristic
43C4×#10624382C2313yes—
44#15668450D411yescharacteristic
45#15668450C2311yescharacteristic
46#15668449—23no—
47#15668449—23no—
48#15668449C2×C413yes—
49#15668449C2×C413yescomplemented
50#15668448—23no—
51#15668448—26no—
52#15668448—23no—
53#15668448—23no—
54#15668448—23no—
55#15668448—23no—
56#15668448C2×C416yescomplemented
57#15668448D413yes—
58#15668448C2313yes—
59C2×Dic362—23no—
60C2×Dic362C2×C413yescomplemented
61Q8×D181—21no—
62Q8×D181C2×C413yescomplemented
63Q8×D181D411yescharacteristic
64C2×C22⋊D181—23no—
65C2×C22⋊D181—26no—
66D4×D181—23no—
67D4×D181—23no—
68D4×D181C2×C413yescomplemented
69D4×D181D413yes—
70C22×Dic181—23no—
71C2×C4×D181—23no—
72C2×C4×D181C2313yes—
73#41×C181—21no—
74#41×C181C2×C411yescharacteristic
75#41×C181C2×C413yescomplemented
76C2×D4×C181—23no—
77C2×D4×C181C2×C413yescomplemented
78C2×#10624382—21no—
79C2×#10624382—23no—
80C2×#10624382—46no—
81#13089685—23no—
82#13089685—21no—
83C22×D181—23no—
84C22×D181C2×D413yes—
85Dic362—23no—
86Dic362—23no—
87Dic362C22⋊C413yes—
88Dic362C22×C413yes—
89D724—21no—
90D724C22⋊C413yes—
91D724C22×C411yescharacteristic
92C22⋊D181—23no—
93C22⋊D181—26no—
94C22⋊D181—23no—
95C22⋊D181—23no—
96C22⋊D181C22⋊C416yes—
97C2×Dic181—23no—
98C2×Dic181—26no—
99C2×Dic181—23no—
100C2×Dic181C2×D413yes—
101C4×D181—23no—
102C4×D181—23no—
103C4×D181—21no—
104C4×D181C22×C413yes—
105C4×D181C2×D413yes—
106C4×D181C2×D413yes—
107C4×D181C2411yescharacteristic
108C23×C181—23no—
109Q8×C181C22⋊C411yescharacteristic
110Q8×C181C22×C411yescharacteristic
111D4×C181—23no—
112D4×C181—23no—
113D4×C181C22⋊C413yes—
114D4×C181C22×C413yes—
115C2×C4×C181—23no—
116C2×C4×C181C22×C413yes—
117C22×C181—23no—
118C22×C181—83no—
119C22×C181C2×C22⋊C413yes—
120C2×D181—21no—
121C2×D181—43no—
122C2×D181—83no—
123C2×D181C2×C22⋊C413yes—
124C2×D181C22×D411yescharacteristic
125#10624382—22no—
126#10624382—43no—
127C4×C181—21no—
128C4×C181C2×C22⋊C413yes—
129C4×C181C23×C411yescharacteristic
130Dic181—23no—
131Dic181C2×C22⋊C413yes—
132Dic181C22×D411yescharacteristic
133C2×C181—23no—
134C2×C181—43no—
135C2×C181C22×C22⋊C411yescharacteristic
136D181—21no—
137D181—23no—
138D181—41no—
139C181#290111yescharacteristic, complemented, Sylow
140#2901—1811nomaximal, Sylow
141#585—1811no—
142#568—1811no—
143C4×#41—1811no—
144#421—1813no—
145#421—1813no—
146C2×#104—1813no—
147C2×#104—1813no—
1482−1+4—3621no—
1492−1+4—1811no—
1502−1+4—1813no—
151H(5,𝔽2)—3623no—
152H(5,𝔽2)—1813no—
153H(5,𝔽2)#1062438211yescharacteristic, complemented
154C2×#41—3623no—
155C2×#41—3621no—
156C2×#41—1813no—
157C2×#41—1813no—
158C2×#41—1811no—
159#131—3623no—
160#131—3621no—
161C2×M16—1813no—
162C2×M16—1813no—
163C4×Q8—3621no—
164C4×D4—3623no—
165#117—1813no—
166C2×C42—1813no—
167#104—1813no—
168#104—1811no—
169#104—1816no—
170#104—1813no—
171#104—1813no—
172#41—21no—
173#41—3623no—
174#41—3623no—
175#41—3623no—
176#41—3626no—
177#41—3623no—
178#41—3623no—
179#41—3623no—
180#41—3623no—
181#41—1811no—
182#41—1813no—
183#41—1816no—
184#41—1813no—
185#41—1811no—
186#41—1813no—
187#41—1813no—
188#41C2×#1062438211yescharacteristic
189#41C2×#1062438213yescomplemented
190C2×Q8—3623no—
191C2×Q8—3621no—
192C2×Q8—1813no—
193C2×Q8—1813no—
194C2×Q8—1811no—
195C2×D4—23no—
196C2×D4—3623no—
197C2×D4—3623no—
198C2×D4—3623no—
199C2×D4—3626no—
200C2×D4—1813no—
201C2×D4—1813no—
202C2×D4C2×#1062438213yescomplemented
203C22×C4—3623no—
204C22×C4—3623no—
205C22×C4—3623no—
206C22×C4—1813no—
207M16—3623no—
208M16—3623no—
209M16—1813no—
210M16—1811no—
211C2×C8—3623no—
212C2×C8—3623no—
213C4⋊C4—3623no—
214C22⋊C4—3623no—
215C42—3623no—
216C42—1811no—
217C42—1813no—
218C23—23no—
219C23—3623no—
220C23—1813no—
221Q8—3623no—
222Q8—3623no—
223Q8—1813no—
224Q8—1813no—
225Q8C22⋊#1062438211yescharacteristic
226Q8C22×#1062438211yescharacteristic
227D4—23no—
228D4—23no—
229D4—3621no—
230D4—3623no—
231D4—3626no—
232D4—3623no—
233D4—3623no—
234D4—1816no—
235D4—1813no—
236D4—1811no—
237D4C22⋊#1062438213yes—
238D4C22×#1062438213yes—
239C2×C4—23no—
240C2×C4—3623no—
241C2×C4—3621no—
242C2×C4—3621no—
243C2×C4—3623no—
244C2×C4—3623no—
245C2×C4—3626no—
246C2×C4—3623no—
247C2×C4—3623no—
248C2×C4—1813no—
249C2×C4—1813no—
250C2×C4—1813no—
251C2×C4—1811no—
252C2×C4—1813no—
253C2×C4—7246no—
254C2×C4C22×#1062438213yes—
255C8—3623no—
256C8—3621no—
257C22—23no—
258C22—83no—
259C22—3621no—
260C22—14483no—
261C22—1811no—
262C22—1813no—
263C22—7243no—
264C22C2×C22⋊#1062438213yes—
265C4—21no—
266C4—3622no—
267C4—3623no—
268C4—1811no—
269C4—1813no—
270C4—7243no—
271C4C2×C22⋊#1062438213yes—
272C4C23×#1062438211yescharacteristic
273C2—23no—
274C2—43no—
275C2—3621no—
276C2—3623no—
277C2—7241no—
278C2C22×C22⋊#1062438211yescharacteristic
subgroupG/H
centerC2C22×C22⋊#10624382
commutatorC4×C181C23×C4
FrattiniC4C23×#10624382
FittingH(5,𝔽2)×C181C4
socleC2×C181C22×C22⋊C4
prime radical#26042114C2
cosocleC4×D181C24
hypercenterH(5,𝔽2)#10624382
hypocenterC181#2901
LCSG ▷ C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ #41 ◁ H(5,𝔽2)

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #26042358C2semidirect
2. #41×#10624382C2semidirect
3. #26042277C2semidirect
4. #26042276C2semidirect
5. D181⋊#104C2semidirect
6. C2×#19153880C2semidirect
10. C2×#15668459C22semidirect
11. D181⋊M16C22semidirect
14. #19153895C22semidirect
15. C2×C4×#10624382C22semidirect
16. #19153881C22semidirect
17. #19153881C22semidirect
18. #19153880C22semidirect
19. #19153880C22semidirect
20. #19153880C22semidirect
22. #19153863C4semidirect
23. #19153863C4semidirect
25. #19153861C4semidirect
31. H(5,𝔽2)×C181C4semidirect
49. #15668449C2×C4semidirect
56. #15668448C2×C4semidirect
60. C2×Dic362C2×C4semidirect
62. Q8×D181C2×C4semidirect
68. D4×D181C2×C4semidirect
75. #41×C181C2×C4semidirect
77. C2×D4×C181C2×C4semidirect
139. C181#2901semidirect
153. H(5,𝔽2)#10624382semidirect
189. #41C2×#10624382semidirect
202. C2×D4C2×#10624382semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#260423582
#41×#106243822
#260422772
#260422762
D181⋊#1042
C2×#191538802
#260421142
#2901181